The impact of ambiguity and prudence on prevention decisions

Abstract

Most decisions concerning (self-)insurance and self-protection have to be taken in situations in which (a) the effort exerted precedes the moment uncertainty realizes, and (b) the probabilities of future states of the world are not perfectly known. By integrating these two characteristics in a simple theoretical framework, this paper derives plausible conditions under which ambiguity aversion raises the demand for (self-)insurance and self-protection. In particular, it is shown that in most usual situations where the level of ambiguity does not increase with the level of effort, a simple condition of ambiguity prudence known as decreasing absolute ambiguity aversion (DAAA) is sufficient to give a clear and positive answer to the question: Does ambiguity aversion raise the optimal level of effort?

Keywords

JEL Classification

Notes

Acknowledgments

The author thanks David Alary, Louis Eeckhoudt, Renaud Foucart, Christian Gollier, François Salanié, Nicolas Treich, and Philippe Weil for helpful comments and discussions. The research leading to this paper received funding from the FRS-FNRS and from the European Union Seventh Framework Programme FP7/2007-2013 under Grant agreement no. 308329 (ADVANCE).

Appendix

Proof of Proposition 4

This proof is based on the following Lemma, that can be found in Gollier (2001).21

Self-protection\((w_{2}^b(e)=w_{2}^b\)for all levels ofe): in this case, the proof is similar but \(U_{j\theta }\) is now given by \(U_{j\theta }=p(e_j,\theta )\mathrm {E} u({\tilde{w}}_{2}^b)+\left[ 1-p(e_j,\theta )\right] \mathrm {E} u({\tilde{w}}_{2}^g)\), and we exploit the convexity of \(p(e,\theta )\) in e to obtain \(\lambda _1 U_{1\theta }+\lambda _2 U_{2\theta }\le U_{\lambda \theta }\). \(\square \)

Proof of Proposition 7

1.

Imagine that the set of risky second period wealth in the bad states of the world can be ranked according to FSD:

Given that \(u'>0\), it is, therefore, clear that \(U(e^*,\theta )=p\mathrm {E} u({\tilde{w}}_2^b(e^*,\theta ))+(1-p) \mathrm {E} u({\tilde{w}}_2^g)\) is decreasing in \(\theta \), and that \(U_e(e^*,\theta )=\mathrm {E} g({\tilde{w}}_2^b(e^*,\theta ))\) will be increasing in \(\theta \) if and only if g is decreasing. Now remark that this will be the case if, by ordering (without loss of generality) the different bad states such that \(w_{2,s}^b\ge w_{2,s+1}^b\), we have:

for all \(\theta \), and all s. It is then easy to see that inequality (16) will be satisfied if \(u''\le 0\) and \(\frac{\partial w_{2,s}^b(e^*,\theta )}{\partial e}\le \frac{\partial w_{2,s+1}^b(e^*,\theta )}{\partial e}\), meaning that the marginal benefit of self-insurance is higher for higher losses.

2.

Imagine that the set of risky second period wealth in the bad states of the world can be ranked according to SSD:

Given that \(u'>0\) and \(u''>0\) under risk aversion, we know that \(U(e^*,\theta )=p\mathrm {E} u({\tilde{w}}_2^b(e^*,\theta ))+(1-p) \mathrm {E} u({\tilde{w}}_2^g)\) is decreasing in \(\theta \). Similarly, \(U_e(e^*,\theta )=\mathrm {E} g({\tilde{w}}_2^b(e^*,\theta ))\) will be increasing in \(\theta \) if and only if \(g'\le 0\) and \(g''>0\). Since we know that \(g(w_{2,s}^b(e^*,\theta ))= u'(w_{2,s}^b(e^*,\theta ))\frac{\partial w_{2,s}^b(e^*,\theta )}{\partial e}\), this will be the case if

for all \(\theta \), all \(i,j\in s\), and all \(\lambda \in [0,1]\). It is then easy to see that inequality (19) is satisfied with an equality if \(u''= 0\), and with a strict inequality if \(\frac{\partial w_{2,i}^b(e^*,\theta )}{\partial e}= \frac{\partial w_{2,j}^b(e^*,\theta )}{\partial e}\) and the agents exhibits risk prudence: \(u'''>0\). More generally, the risk prudence property implies that: